Horizontal and Vertical Curves and Sight Distance: Method

Horizontal and Vertical Curves and Sight Distance

This page describes the seven computations of the tool: the simple circular curve and its staking table; compound and reverse curves; the clothoid transition spiral; superelevation and the minimum radius for a design speed; the parabolic vertical curve with its sight-distance criteria and the cut or fill of a ground line; and stopping sight distance with the horizontal sightline offset. The surveying methods are those of the public-domain manuals of the U.S. Army (FM 5-233, Construction Surveying, 1985) and the U.S. Navy (NAVEDTRA 14337, Engineering Aid 2, 1994); the design criteria (stopping sight distance, crest and sag curve lengths, the point-mass curve equation, the side-friction factors) are the AASHTO Green Book relations as reprinted in public FHWA, BLM and Army documents, used here as mathematics with their constants as editable inputs. The Green Book itself is cited only; no table of it is reproduced.

Stations and conventions

Distances along the centre line are measured from the origin of the survey in full stations of 100 ft (USCS) and written as the full station plus the remainder: 1,250 ft is station 12+50, 982.05 ft is 9+82.05. The tool takes every station as a plain distance (type 1250) and prints the +00 form. In SI the full station is taken as 100 m and the label is written kilometre + metres (1+250.00); the degree of curve is then the central angle of a 100 m arc or chord. Angles are entered in degrees and printed in degrees and decimal minutes (12°30.00′). Grades are in percent, rising positive in the direction of stationing. Design speeds are plain numbers, mph in USCS and km/h in SI; the unit switch converts lengths and heights but not the speed, the time or the grades. Every curve is drawn turning left; the sense in the field is set by the alignment.

Simple circular curve

The degree of curve D is the central angle subtended by one full station: by the arc definition (highways) R = 100 (180/π)/D = 5,729.58/D ft, by the chord definition (railroads, FM 5-233) R = 50/sin(D/2) ft. With the intersection angle Δ (= I, the deflection of the tangents at the PI, equal to the central angle): tangent distance T = R tan(Δ/2), length L = R Δ (arc definition; 100 Δ/D along the chords under the chord definition), external distance E = R (1/cos(Δ/2) − 1) = T tan(Δ/4), middle ordinate M = R (1 − cos(Δ/2)) = E cos(Δ/2), long chord LC = 2R sin(Δ/2); PC = PI − T and PT = PC + L (FM 5-233 pages 3-4 to 3-8, NAVEDTRA 14337 pages 11-5 to 11-9). Staking: with the instrument at the PC sighted along the back tangent, the deflection angle to a point at arc distance C from the previous stake is d = (D/2)(C/100) (= 0.3 C D minutes), exact under the arc definition because the inscribed angle is half the central angle C/R; under the chord definition it is the usual approximation. The deflections accumulate to Δ/2 at the PT; the true chord to tape between stakes is 2R sin d. The usual chord is 100 ft up to D = 3°, 50 ft to 8°, 25 ft to 16° and 10 ft beyond (FM 5-233 page 3-5), or any entered interval.

Compound curve

Two simple curves of radii R₁, R₂ turning the same way meet at the point of compound curvature (PCC), the PT of the first being the PC of the second; I = Δ₁ + Δ₂. FM 5-233 (pages 3-18 to 3-20) gives two layouts. Between successive tangents (the main PI and both curves known): T₁ = R₁ tan(Δ₁/2), T₂ = R₂ tan(Δ₂/2), AB = T₁ + T₂ is the common tangent between PI₁ and PI₂, the triangle PI₁–PI₂–PI has the angle C = 180° − I at the PI, so by the sine rule AC = AB sin Δ₂/sin C and BC = AB sin Δ₁/sin C; PI₁ = PI − AC, PC = PI₁ − T₁, PCC = PC + L₁, PI₂ = PCC + T₂, PT = PCC + L₂; the long and short tangents of the whole curve are AC + T₁ and BC + T₂. Between successive PIs (the first curve and the distance AB known): T₂ = AB − T₁ and R₂ = T₂/tan(Δ₂/2), the second degree of curve carried to five places because the tangent must be held exact. The manual advises that the two degrees should not differ by more than 3°. Each arc is staked as a simple curve, the second from the PCC with the instrument oriented on PI₂. The tool's default is the NAVEDTRA 14337 Figure 11-22 problem (Δ₁ 60°, Δ₂ 45°, D₁ 7°, D₂ 12° arc, PI 28+75), whose printed answers (PC 19+11.71, PI₁ 23+84.28, PCC 27+68.85, PI₂ 29+66.62, PT 31+43.85; 12° curve deflections 1°52.1′, 2°37.9′, 6° and chords 31.13, 43.84, 99.81 ft) the page reproduces.

Reverse curve

Two arcs turning opposite ways meet at the point of reverse curvature (PRC). Parallel tangents a distance p apart (FM 5-233 Figure 3-13): with the PRC m₁ from the first tangent (p/2 when the arcs are equal), cos I₁ = (R₁ − m₁)/R₁ and L₁ = R₁ sin I₁ is the distance along the first tangent from the PC to the foot of the PRC; likewise for the second arc with m₂ = p − m₁; the PC and PT are located by measuring L₁ and L₂. The arcs are tangent at the PRC only when the two central angles are equal (m₁/R₁ = m₂/R₂); the page reports both angles and the offset that makes them equal. The manual's example (5° chord-definition curves, p = 225 ft) gives R = 1,146.29 ft, cos I₁ = 0.901857, I₁ = 25°36′, L₁ = 495.30 ft. Diverging tangents (Figure 3-14): the forward tangent deflects I at the PI and the PT is fixed T_s back from the PI along it; the perpendicular to the forward tangent at the PT carries the second centre and meets the back tangent at m = T_s tan I from the PT and L = T_s/cos I from the PI; n = (R₂ − m) sin I and p = (R₂ − m) cos I locate the second centre, the centres are R₁ + R₂ apart so cos I₁ = (R₁ + p)/(R₁ + R₂), I₂ = I₁ + I, g = (R₁ + R₂) sin I₁, and the PC is T_L = g + n + L from the PI. The manual's example (I = 41°, T_s = 550 ft, both arcs 5°) gives m 478.11, L 728.76, n 438.37, p 504.28, I₁ 43°57′, g 1,591.12 and T_L 2,758.25 ft; the tool's vector closure check reproduces the PT from the PC through both arcs to 10⁻¹³ ft. Reverse curves cannot be superelevated at the PRC and are kept off high-speed roads.

Spiral transition

The clothoid's curvature grows linearly with the distance l from the TS, so the angle turned is θ = l²/(2 R L_s) and the coordinates along and across the TS tangent are the Fresnel integrals, evaluated by their series x = l (1 − θ²/10 + θ⁴/216 − θ⁶/9,360 + …), y = l (θ/3 − θ³/42 + θ⁵/1,320 − …), which the tool sums to fourteen terms (10⁻¹³ against scipy's Fresnel functions). At the SC, θ_s = L_s/(2R), written D L_s/200 with the degree of curve in the railroad convention (FM 5-233 page 3-27; the two agree under the arc definition and differ in the fourth figure under the chord definition, which the "θ_s by" select reproduces). The elements follow by geometry: the throw p = Y − R(1 − cos θ_s) (the manual's o, the offset of the shifted circular curve from the tangent), k = X − R sin θ_s (its Z, the abscissa of the shifted PC), long tangent LT = X − Y/tan θ_s, short tangent ST = Y/sin θ_s, long chord C = √(X² + Y²) with the deflection A = atan(Y/X) to the SC (≈ θ_s/3; the manual's A = θ_s/3 − 0.00297 θ_s³ seconds) and B = θ_s − A at the SC; T_s = (R + p) tan(I/2) + k, E_s = (R + p)/cos(I/2) − R; the circular arc keeps Δ_c = I − 2θ_s and L_c = R Δ_c; TS = PI − T_s, SC = TS + L_s, CS = SC + L_c, ST = CS + L_s. The FM 5-233 ten-chord forms (A as above, C = L_s(cos 0.3θ_s + 0.004 exsec ¾θ_s), X = C cos A, Y = C sin A) are printed beside the exact values. Staking: the spiral is set out in n equal chords from the TS; the deflection to the point at l is atan(y/x), which the manuals take as θ/3 so that a_n = n² a₁ with a₁ = 10 k s² minutes (k the increase in degree of curve per station, s the chord in stations); the circular arc is staked from the SC with d = (D/2)(C/100) after backsighting the TS with B on the circle, and the deflections reach Δ_c/2 at the CS. The manual's example (D = 4° chord, I = 24°10′, L_s = 250 ft, PI 42+61.70, 60 mph) gives Δ_s 5°, X 249.81, Y 7.27, o 1.81, Z 124.94, T_s 432.04, L_c 354.17, TS 38+29.66, SC 40+79.66, CS 44+33.83, ST 46+83.83, spiral deflections 0°01′, 0°04′ … 1°40′ and arc deflections 0°24.4′ … 7°05′, which the tool reproduces with the "D L_s/200" option. Spiral length: FM 5-233 page 3-28 gives the highway minimum L_s = 1.6 V³/R (ft, mph; its Table 3-1 below rounds up to 50 ft) and the railroad rule L_s = 1.17 E V (E the superelevation in inches); Shortt's form L = 3.15 V³/(R C) with the rate of change of lateral acceleration C (1 to 3 ft/s³; metric 0.0214 V³/(R C)) is printed beside it.

FM 5-233 Table 3-1 (page 3-29), recommended superelevation e (ft/ft) and minimum transition length L_s (ft) by degree of curve and design speed, e_max 0.10; transcribed from the scan (the 2°30′ / 30 mph cell is faint and read as .01):

Superelevation and minimum radius

A vehicle of speed V on a curve of radius R banked at the rate e needs the centripetal acceleration V²/R from the bank and the side friction f: 0.01 e + f = V²/(15 R) with V in mph and R in ft (V²/(127 R) in km/h and m), the point-mass equation printed as equation 2 of the FHWA Speed Concepts guide (FHWA-SA-10-001, page 14). With the maximum rate of superelevation e_max (a policy value, 4 to 12 %) and the maximum design side-friction factor f_max at the design speed, the minimum radius is R_min = V²/(15 (0.01 e_max + f_max)) and the maximum degree of curve 5,729.58/R_min. The side-friction factors offered by the speed select are read from Figure 5 of the same guide (the 2004 Green Book design curve as reprinted by FHWA): 0.32 at 15 mph falling to 0.08 at 80 mph; the BLM Road Design Handbook (H-9113-1, Illustration 3) uses 0.16 for paved and 0.09 to 0.12 for unpaved low-volume roads. For an entered curve the page gives the friction demanded at the design speed with the design rate, the balanced rate at which f = 0, the rate needed with the full f_max, the speed at which f_max is reached and the hands-off speed. The runoff length over which the outer edge rotates from level to the full rate is L_r = w n₁ e_d b_w/Δ with the lane width w, n₁ lanes rotated, the adjustment b_w (1, 0.75, 0.67 for 1, 2, 3 lanes) and the maximum relative gradient Δ between the rotated edge and the axis of rotation (0.50 % at 50 mph; the BLM Illustration 9 runoff lengths imply 1.00, 0.89, 0.80, 0.67, 0.57 and 0.50 % at 10 to 50 mph; the Green Book Table 3-15 is the policy source); the tangent runout from the normal crown is L_t = (e_NC/e_d) L_r. The distribution of e between R_min and the no-superelevation radius (AASHTO Method 5) is not implemented; its tables are the Green Book's.

Vertical curve

The equal-tangent parabola of length L joins the grade g₁ to g₂ through the PVI with the middle offset e = L (g₂ − g₁)/8 (positive for a sag, the curve above the PVI); the elevation at the distance x from the PVC is the tangent elevation plus (x/l)² e, measured from the PVC on the back tangent and from the PVT on the forward tangent, and the second differences at equal intervals are constant (FM 5-233 pages 3-35 to 3-38, NAVEDTRA 14337 pages 11-13 to 11-19). An unsymmetrical curve with tangents l₁, l₂ is two parabolas sharing the grade at the PVI with e = l₁ l₂ (g₂ − g₁)/(2 (l₁ + l₂)). The rate of vertical curvature K = L/|A|, A the algebraic difference of the grades in percent, is the length per percent of grade change; the high or low point lies K g₁ from the PVC (on the branch of the flatter grade for an unsymmetrical curve) and exists only when the grades differ in sign. The NAVEDTRA chapter questions Q3 (400 ft, +3.2 / −1.6 %, PVI 14+00 at 131.20) and Q4 (unsymmetrical 400 / 200 ft, −3 / −8 %, PVI 15+00 at 640) and the FM 5-233 Figure 3-20 sag are reproduced by the test to the printed digits.

Sight distance

Stopping sight distance is the distance travelled during the brake perception-reaction time plus the braking distance: S = 1.47 V t + V²/(30 ((a/32.2) + G)) in ft with V in mph, a in ft/s² and the grade G as a decimal, downgrade negative (S = 0.278 V t + V²/(254 ((a/9.81) + G)) in m, km/h, m/s²). The design assumptions stated in FHWA-SA-10-001 (page 10) are t = 2.5 s, a = 11.2 ft/s² (3.4 m/s², 0.35 g), eye height 3.5 ft and object height 2.0 ft; the AASHTO design values it reprints in Table 2 (80 ft at 15 mph to 910 ft at 80 mph) are the equation rounded up to 5 ft, except 820 ft at 75 mph, 5 ft above the rule (814.7 ft computed). On a crest the sight line grazes the curve: with C = 200 (√h₁ + √h₂)² (= 100 (√(2h₁) + √(2h₂))², 2,158 ft with 3.5 / 2.0 ft, 2,800 ft with the 3.5 ft passing object, 658 m with 1.08 / 0.60 m) the minimum length for the sight distance S is L = A S²/C when S < L and L = 2S − C/A when S > L (the first form is printed in BLM H-9113-1 Illustration 5 with its older 3.75 / 0.50 ft heights); the available sight distance of a given curve follows by inverting the same two relations. On a sag the night-time criterion is the headlight beam from the height h₃ inclined β upward: L = A S²/(200 (h₃ + S tan β)) when S < L and L = 2S − 200 (h₃ + S tan β)/A when S > L, which with h₃ = 2 ft and β = 1° is the A S²/(400 + 3.5 S) of BLM Illustration 6; the comfort criterion L = A V²/46.5 (ft, mph; 395 in m, km/h) gives about half the headlight length and governs only with fixed lighting. The tool also traces the available sight distance from every driver station numerically on the whole profile (tangent, parabola, tangent): on a crest the longest sight line to the object that stays above the road, on a sag the point where the beam, inclined β above the vehicle's own grade, meets the road; the minimum of the crest envelope equals the closed form when the governing line lies within the curve, and the sag reach at the PVC equals the closed form (the AASHTO criterion puts the vehicle at the PVC; beyond it the beam meets the rising grade sooner, which the criterion does not count). A K above 167 ft per percent (51 m) leaves less than 0.3 % of grade within 50 ft of the vertex and calls for attention to drainage. Decision and passing sight distances are entered from the agency's table and compared with the available distance; the ratio of the required to the available distance and its margin are printed through the site's margin box with "AASHTO" named as the source of the criterion.

Horizontal sightline offset: on a curve of radius R the sight line of length S between a driver and an object on the centre line of the inside lane is a chord whose middle ordinate is the clear offset needed to an obstruction, M = R (1 − cos(28.65 S/R)) (28.65 S/R being the half central angle in degrees, S/(2R) in radians), and an obstruction at M allows S = (R/28.65) acos((R − M)/R); both hold when the sight distance lies within the curve. When the curve is shorter than S the tool finds the offset numerically as the largest distance between the sight line and the lane line over every driver position from S before the PC to the PT. TM 5-822-2 (page 3-13) requires the clear sight distance along the inside lane to equal the minimum stopping sight distance, and twice that on single-lane roads; the FLH PDDM (Exhibit 9.3-J) adds the shoulder, ditch and slope offsets to the lane centre.

Cut and fill

Entered ground elevations at stations are compared with the design profile, the curve inside PVC–PVT and the tangent grades outside; ground above the design is cut, below it fill. The heights are the centre-line cut or fill of each cross section; the Earthwork Volumes and Mass Haul tool takes the end areas from there to the volumes and the mass diagram.

Sources and copyright

FM 5-233 and NAVEDTRA 14337 are U.S. government works; their simple, compound, reverse, spiral and vertical-curve examples are the acceptance cases of the test and three of them are the tool's defaults. FM 5-233 page ii notes that the manual contains copyrighted material reproduced by permission of the American Railway Engineering Association: that material is the spiral tables of its appendix, from which nothing is taken; the spiral is computed from the clothoid series and the manual's own formulas on pages 3-27 and 3-28 are quoted as mathematics. The FHWA Speed Concepts guide (2009), the FHWA Mitigation Strategies for Design Exceptions (2007), the FLH Project Development and Design Manual chapter 9 (2012), the BLM Road Design Handbook (2011) and TM 5-822-2 (1987) are government works used for the design relations and the sample values they print; the AASHTO Green Book and NCHRP Report 400 are cited only. The default horizontal curve (R 1,000 ft, Δ 30°, PI 12+50) and crest curve (600 ft, +3 % to −2 %, PVI 25+00 at 500.00, 50 mph) are illustrative and are checked against the closed forms and an independent numpy / scipy script; the ground elevations of the default profile are illustrative too.